Rotational Symmetry

An object has rotational symmetry if it looks the same after a turn of some size.

The number of times that the object matches up to itself during a full turn is called the order of the rotational symmetry.

For example, the star below has rotational symmetry of order 5. It matches up with itself 5 times as it passes through a full turn. I have added a small circle to one point of the star so that you can see it moving around as the star rotates:

Some Objects and Geometric Shapes with Rotational Symmetry

Order \(2\)

Order \(3\)

Order \(4\)

Order \(5\)

The stars on the New Zealand flag have rotational symmetry of order \(5\).

Order \(6\)

Order \(7\)

All but one star on the Australian flag has rotational symmetry of order \(7\).

Order \(8\)

Order \(9\)

Any regular polygon has rotational symmetry of order equal to its number of sides.

Rotational Symmetry of Order 1?

Just think what rotational symmetry of order \(1\) would mean. It wouldn't be very interesting, would it? It just says that the object matches up to itself after one whole turn. So, rotational symmetry of order one really means no rotational symmetry at all!

Conclusion

Have a look around and see if you can spot any objects with rotational symmetry. Can you work out the orders?

Another name for rotational symmetry of order \(2\) is point symmetry.

Have you thought about circles and rotational symmetry? I guess a circle has rotational symmetry of infinite order, because it will match up to itself after turning through any angle at all.